Ferrimagnetic oscillator magnetometer

ABSTRACT

Ferrimagnetic oscillator magnetometers do not use lasers to stimulate fluorescence emission from defect centers in solid-state hosts (e.g., nitrogen vacancies in diamonds). Instead, in a ferrimagnetic oscillator magnetometer, the applied magnetic field shifts the resonance of entangled electronic spins in a ferrimagnetic crystal. These spins are entangled and can have an ensemble resonance linewidth of approximately 370 kHz to 10 MHz. The resonance shift produces microwave sidebands with amplitudes proportional to the magnetic field strength at frequencies proportional to the magnetic field oscillation frequency. These sidebands can be coherently averaged, digitized, and coherently processed, yielding magnetic field measurements with sensitivities possibly approaching the spin projection limit of 1 attotesla/√{square root over (Hz)}. The encoding of magnetic signals in frequency rather than amplitude relaxes or removes otherwise stringent requires on the digitizer.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims the priority benefit, under 35 U.S.C. 119(e), ofU.S. Application No. 63/050,832, filed on Jul. 12, 2020, which isincorporated herein by reference in its entirety for all purposes.

GOVERNMENT SUPPORT

This invention was made with government support under FA8702-15-D-0001awarded by the U.S. Air Force. The government has certain rights in theinvention.

BACKGROUND

In recent years, tremendous experimental effort has been dedicated tothe development of quantum sensors employing unpaired electron spinsembedded in solid-state crystals. These solid-state sensors employelectron paramagnetic resonances to offer measurement precision andaccuracy comparable to their atomic counterparts, with substantialadvantages such as sensor size, compatibility over a wide range ofambient conditions, and a rigid crystal lattice providing fixed sensingaxes. The most-developed solid-state quantum sensing platform usesnegatively charged nitrogen-vacancy (NV) centers in diamond as sensitiveprobes of magnetic field. Such sensors have been employed for detectionor imaging of biological targets, single proteins, nuclear magneticresonance (NMR) species, individual spins, and condensed matterphenomena.

Although NV centers in diamond offer important advantages as a quantumsensing platform, including some of the longest coherence times of anyparamagnetic defect in a solid-state material, several barriers exist torealizing their full capabilities. Despite sustained effort,high-fidelity state readout of NV ensemble sensors remains challenging.Additionally, the high-intensity optical light used to initialize NVsinto a single quantum state (e.g., 100 kW/cm²) can result in bothprohibitively large optical powers and thermal dissipation challengesfor mm³-scale ensembles. Finally, various technical obstacles, such aschallenges in growing optimized diamond material and miniaturizingoptical and microwave components, should be addressed before NV⁻ baseddevices can be widely employed outside of laboratory environments.

SUMMARY

Though recent efforts have focused on optically active paramagneticdefects, ferrimagnetic and ferromagnetic materials can offer distinctadvantages for quantum sensors. In a ferromagnetic material, adjacentspin magnetic moments are aligned parallel to each other, resulting inspontaneous magnetization even in the absence of an ambient magneticfield, unlike in paramagnetic materials. In a ferrimagnetic material,adjacent spin magnetic moments are aligned anti-parallel to each other,but do not completely cancel each other. This also results in aspontaneous magnetization even in the absence of an ambient magneticfield. Ferrimagnetic and ferromagnetic materials can provide much higherspin densities than their paramagnetic counterparts (e.g., ˜10²² cm⁻³vs.˜10¹⁶-10¹⁹ cm⁻³) while the strong coupling of the exchangeinteraction mitigates the dipolar resonance broadening often observed inhigh-defect-density paramagnetic materials. In addition, ferromagneticor ferrimagnetic materials can be passively initialized into the desiredquantum state by application of a bias magnetic field rather thanactively initialized with light.

A magnetometer can use a ferrimagnetic (or ferromagnetic) material as amagnetically sensitive filter component in an oscillator loop. Theferrimagnetic material acts as a notch or passband filter whoseresonance or center frequency varies with the externally appliedmagnetic field. When coupled to a gain component, such as a sustainingamplifier, in a transmission geometry, the ferrimagnetic filter cansustain an oscillating voltage at the resonance frequency. Theamplitude, phase, and frequency of the external magnetic field sensed bythe ferro/ferrimagnetic oscillator magnetometer can be encoded in thefrequency modulation of the oscillator output voltage, in an errorsignal fed back to the oscillator to keep the oscillator locked to theferrimagnetic resonance, or in a combination of both. Other geometriesare also possible; for example, the oscillator can be implemented as aPound-Galani oscillator where reflection from the ferromagnetic materialis used to lock the oscillator frequency to the ferro/ferrimagneticresonance and transmission through the ferro/ferrimagnetic material isused to sustain the oscillation.

A ferrimagnetic oscillator magnetometer can achieve a sensitivity of aslow as 140 fT/√{square root over (Hz)}, with sensitivity below 300fT/√{square root over (Hz)} over a broad range of frequencies (e.g.,about 10 kHz to about 1 MHz). In addition, the sensor head can be small(e.g., the sensor volume can be about one cubic inch or less), rugged,robust to vibration, and simple to make. It uses microwave componentsand can be fabricated lithographically, with the ferrimagnetic crystalformed using inexpensive sputtering (ferrimagnetic Yttrium Iron Garnet(YIG) forms a single crystal when sputtered). It does not require alaser, photodetector, or external microwave source, so its powerconsumption can be much lower than other quantum magnetometers ofcomparable sensitivity. It can operate at room temperature and withoutcalibration.

An example magnetometer may include a ferrimagnetic crystal, asustaining amplifier in electrical communication with the ferrimagneticcrystal, and a digitizer in electrical communication with the sustainingamplifier and/or the ferrimagnetic crystal. The ferrimagnetic crystalincludes an ensemble of entangled electronic spins with a resonance thatshifts in response to an external magnetic field. In operation, thesustaining amplifier amplifies a microwave signal modulated by a shiftin the resonance of the ensemble of entangled electronic spins. And thedigitizer digitizes the microwave signal.

The ferrimagnetic crystal, which may include the entangled electronicspins, and the sustaining amplifier, which may include a bipolarjunction transistor, can be connected in a transmission geometry or areflection geometry. The magnetometer may also include a bandpassfilter, in electromagnetic communication with an input of the sustainingamplifier, to filter the microwave signal. And it can include a biasmagnet, in electromagnetic communication with the ferrimagnetic crystal,to apply a bias magnetic field to the ensemble of entangled electronicspins.

The shift in the resonance can vary linearly with an amplitude of theexternal magnetic field. It can modulate sidebands onto the microwavesignal with amplitudes proportional to an amplitude of the externalmagnetic field at offset frequencies proportional to an oscillationfrequency of the external magnetic field. In this case, the magnetometermay have a sensitivity versus the oscillation frequency of the externalmagnetic field that is substantially constant for f_(c)<f_(m)<f_(L),where f_(m) is the oscillation frequency of the external magnetic field,f_(c) is an observed noise corner of the sustaining amplifier, and f_(L)is the Leeson frequency of the magnetometer.

The magnetometer may also include input and output coupling loops thatare inductively coupled to the ferrimagnetic crystal and couple themicrowave signal into and out of the ferrimagnetic crystal,respectively. The magnetometer may also include a directional couplerwith at least three ports: (1) an input port coupled to the outputcoupling loop, (2) a through port coupled to an input of the sustainingamplifier, and (3) a tap port coupled to the digitizer. Alternatively,or in addition, the magnetometer may include a feedback loop, inelectromagnetic communication with the ferrimagnetic crystal and thesustaining amplifier, to generate and apply an error signal correctingan error between a frequency of the microwave signal and a centerfrequency of the resonance.

A self-sustaining oscillator comprising a ferrimagnetic material thatexhibits a ferrimagnetic resonance and is operably coupled to asustaining amplifier can sense an alternating current (AC) magneticfield as follows. The AC magnetic field is applied to the ferrimagneticmaterial, which may be subject to a bias magnetic field, therebyshifting a center frequency of the ferrimagnetic resonance exhibited bythe ferrimagnetic material and modulating a microwave oscillationsupported by the self-sustaining oscillator. While the AC magnetic fieldis applied to the ferrimagnetic material, the sustaining amplifieramplifies the microwave oscillation transmitted by the ferrimagneticresonance. The amplitude and/or a frequency of the AC magnetic field canbe determined based on the modulation of the microwave oscillation.

Modulating the microwave oscillation can produce sidebands withamplitudes proportional to an amplitude of the external magnetic fieldat frequencies proportional to an oscillation frequency of the externalmagnetic field. If so, the amplitude and/or the frequency of the ACmagnitude field magnetometer can be determined with a sensitivity versusthe oscillation frequency of the external magnetic field that issubstantially constant for f_(c)<f_(m)<f_(L), where f_(m) is theoscillation frequency of the external magnetic field, f_(c) is anobserved noise corner of the sustaining amplifier, and f_(L) is theLeeson frequency of the magnetometer.

Determining the amplitude and/or the frequency of the AC magnetic fieldmay include measuring a real component of the microwave oscillation,reconstructing a complex representation of the microwave oscillationfrom the real component of the microwave oscillation, determining aphase angle of the microwave oscillation as a function of time based onthe complex representation, and determining the AC magnetic field basedon the phase angle. Determining the amplitude and/or the frequency ofthe AC magnetic field can also include coherently averaging a digitalrepresentation of the microwave oscillation.

In some cases, a feedback loop generates an error signal correcting anerror between a frequency of the microwave oscillation and a centerfrequency of the resonance. A servo corrects the error based on theerror signal.

Two ferrimagnetic oscillator magnetometers can be coupled to form agradiometer. The first ferrimagnetic oscillator magnetometer generates afirst signal representing an amplitude, frequency, and phase of anexternal magnetic field at a first location, and the secondferromagnetic oscillator magnetometer generates a second signalrepresenting an amplitude, frequency, and phase of the external magneticfield at a second location. A mixer, which is operably coupled to thefirst and second ferrimagnetic oscillator magnetometers, mixes the firstsignal with the second signal, thereby producing a beat signalrepresenting a gradient of the external magnetic field.

All combinations of the foregoing concepts and additional conceptsdiscussed in greater detail below (provided such concepts are notmutually inconsistent) are part of the inventive subject matterdisclosed herein. In particular, all combinations of claimed subjectmatter appearing at the end of this disclosure are part of the inventivesubject matter disclosed herein. The terminology used herein that alsomay appear in any disclosure incorporated by reference should beaccorded a meaning most consistent with the concepts disclosed herein.

BRIEF DESCRIPTIONS OF THE DRAWINGS

The skilled artisan will understand that the drawings primarily are forillustrative purposes and are not intended to limit the scope of theinventive subject matter described herein. The drawings are notnecessarily to scale; in some instances, various aspects of theinventive subject matter disclosed herein may be shown exaggerated orenlarged in the drawings to facilitate an understanding of differentfeatures. In the drawings, like reference characters generally refer tolike features (e.g., elements that are functionally and/or structurallysimilar).

FIG. 1A shows a ferrimagnetic oscillator magnetometer with afree-running transmission geometry.

FIG. 1B shows an architecture for the digitizer at the output of theferrimagnetic oscillator magnetometer of FIG. 1A.

FIG. 1C shows an equivalent circuit for the YIG sphere and couplingloops in the ferrimagnetic oscillator magnetometer of FIG. 1A. Theequivalent circuit for the YIG sphere can be modelled as a series RLCcircuit with idealized inductive coupling and a non-reciprocal 7 r/2phase shifter to account for the gyrator action of the YIG.

FIG. 2A illustrates how the spins of a ferrimagnetic sphere precess inphase in the presence of a uniform external magnetic field.

FIG. 2B is a plot of the spin precession frequency of the uniformferrimagnetic resonance mode versus applied magnetic field.

FIG. 2C is a plot showing the oscillation frequency of (top) andmagnetic field applied to (bottom) a ferrimagnetic oscillatormagnetometer with a ferrimagnetic resonance employed as a frequencydiscriminator.

FIG. 3A shows a process for recovering the AC magnetic field measuredwith a ferrimagnetic oscillator magnetometer from the oscillator voltagewaveform with simulated data.

FIG. 3B shows a sample AC magnetic field waveform recovered from anoscillator waveform using the process of FIG. 3A.

FIG. 4A shows the single-sideband phase-noise power spectral density ofthe ferrimagnetic oscillator magnetometer of FIG. 1A. Thesingle-sideband phase-noise is −122.5 dBc/Hz and −148.5 dBc/Hz at 10 kHzand 100 kHz offsets from the carrier, respectively.

FIG. 4B shows the predicted and measured frequency response of theferrimagnetic oscillator magnetometer of FIG. 1A to a 150 nT rms ACmagnetic field applied along the 2 axis.

FIG. 4C is a plot of the sensitivity versus frequency of theferrimagnetic oscillator magnetometer of FIG. 1A. The device achieves aminimum sensitivity of 140 T/√Hz to AC magnetic fields of known phase atfrequencies near 100 kHz. The sensitivity is below 300 fT/√Hz over theband from 10 kHz to 1 MHz.

FIG. 5A shows a ferrimagnetic oscillator magnetometer with aPound-Galani architecture.

FIG. 5B shows how the magnetic field amplitude is encoded in thePound-Drever-Hall (PDH) error signal in the ferrimagnetic oscillatormagnetometer of FIG. 5A.

FIG. 6A shows a ferrimagnetic oscillator magnetometer with a reflectionoscillator geometry.

FIG. 6B shows an equivalent circuit for the ferrimagnetic oscillatormagnetometer of FIG. 6A.

FIG. 7A is an exploded view of a ferrimagnetic oscillator magnetometerwith a reflection oscillator geometry as in FIGS. 6A and 6B.

FIG. 7B is a photograph of the ferrimagnetic oscillator magnetometer ofFIG. 7A.

FIG. 7C is a photograph of the inside of the ferro/ferrimagneticoscillator magnetometer of FIG. 7A.

FIG. 7D is a plot of the phase noise performance of the ferrimagneticoscillator magnetometer of FIGS. 7A-7C.

FIG. 8 is a plot of coherently averaged frequency uncertainty vs. numberof samples for the ferrimagnetic oscillator magnetometer of FIGS. 7A-7C.

FIG. 9 illustrates how a ferrimagnetic oscillator magnetometer measuresmagnetic field direction and amplitude.

FIG. 10 shows a ferrimagnetic gradiometer.

DETAILED DESCRIPTION

Quantum sensors based on atomic gases or electron spins in solid-statecrystals operate by localizing a resonance which varies with theparameter of interest. For example, the magnetic field may be deduced bymeasuring the frequency of uniform precession of a sphericalferrimagnetic sample, of the paramagnetic resonance of NVs in diamond,or of Zeeman transitions in an alkali vapor. Several experimentaltechniques have been developed for this task, from continuous-wave (CW)absorption or dispersion measurements to pulsed protocols such as Ramseyor pulsed electron spin resonance (ESR) schemes. In all these methods,external radio-frequency (RF) or optical fields manipulate the system,and the location of the resonance is inferred from the resultingresponse as a function of frequency of the applied field.

It is also possible, however, to employ a measurement architecture wherethe system sustains self-oscillation, generating a microwave (MW) outputthat encodes properties of the system's environment, rather than probingthe system with an external MW field. A self-sustaining oscillatorincludes two main components: (1) a frequency-discriminating element and(2) a gain medium subject to some form of feedback (e.g., arranged in aloop with the frequency-discriminating element).

In the self-sustaining ferrimagnetic (and ferromagnetic) oscillatormagnetometers disclosed here, a ferrimagnetic (or ferromagnetic)material serves as the frequency-discriminating element. Theferrimagnetic material acts as a quantum frequency discriminator, with aquantum mechanical resonance (here, also called a ferrimagneticresonance) provided by an ensemble of entangled electronic spins. Thisferrimagnetic resonance can be employed in a variety of ways to providefrequency discrimination of input fields; for example, the ferrimagneticmaterial can be strongly coupled to a cavity whose transmission andreflection depend on the detuning of the applied field relative to theferrimagnetic resonance. Alternatively, the input and output couplersthat couple the MW fields to the ferrimagnetic material can beorthogonal to each other, with negligible coupling to each other in theabsence of the ferrimagnetic resonance. If the MW fields at the inputand output couplers are each coupled to the quantum mechanicaltransition, the ferrimagnetic material provides a frequency-dependentcoupling between the input and output couplers, where the transmissionis maximized at the resonance frequency and negligible at largedetunings from resonance.

The gain for the self-sustaining, ferrimagnetic oscillator magnetometercan be provided by an ordinary RF amplifier that returns a portion ofits output to the input coupling loop, realizing sustainedself-oscillation (under suitable conditions). No external microwavesignal source is necessary; the RF amplifier emits radiation over a bandencompassing the ferrimagnetic resonance frequency ω_(y) of theferrimagnetic material, which transmits oscillations near theferrimagnetic resonance frequency ω_(y) and block oscillations at otherfrequencies. The transmitted oscillations propagate back to the input ofthe RF amplifier, which amplifies them to produce a self-sustainingoscillation at an oscillation frequency ω_(c) that closely tracks theferrimagnetic resonance frequency ω_(y). Thus, the oscillatorarchitecture eliminates the need for a tunable external RF generator inthe system.

In addition, encoding the signal in frequency rather than in amplitudeconfers several benefits, as frequency measurements can offer greaterdynamic range and linearity compared to amplitude measurements. Dynamicrange is particularly useful for a magnetometer; for example, detectionof a 100 fT signal in Earth's field (about 0.1 mT) usually requires adynamic range of about 10⁹.

Ferrimagnetic oscillator magnetometers are amenable to lithographic-typemanufacturing processes. The ferrimagnetic crystal can be a thin layer(e.g., 30 microns or less) sputtered or otherwise deposited on a flatsurface, such as part of a circuit board or integrated circuit. Theferrimagnetic crystal's high number of spins (e.g., about 5×10¹⁹) almostall of which can be utilized, and narrow resonance linewidth (e.g., downto 370 kHz due to exchange narrowing from the entangled iron electrons)give a lower spin projection limit than any other sensor. The signal canbe coherently averaged in the analog domain with the sensor or in thedigital domain after digitization.

At steady state, a ferrimagnetic oscillator magnetometer satisfies theBarkhausen stability criterion, which is a mathematical condition todetermine when a linear electronic circuit will oscillate. To satisfythe Barkhausen stability criterion, the oscillation frequency shouldcorrespond to an integer number of wavelengths around the oscillatorloop, with unity total gain. This criterion, under reasonableassumptions, results in a phase-noise spectral density described byLeeson's equation,

$\begin{matrix}{{{\mathcal{L}^{\frac{1}{2}}\left( f_{m} \right)} = \sqrt{{{\frac{1}{2}\left\lbrack {\frac{f_{L}^{2}}{f_{m}^{2}} + 1} \right\rbrack}\left\lbrack {\frac{f_{c}}{f_{m}} + 1} \right\rbrack}\left\lbrack \frac{Fk_{B}T}{P_{s}} \right\rbrack}},} & (1)\end{matrix}$where

^(1/2)(f_(m)) is the single-sideband phase-noise spectral density atoffset frequency f_(m) from the carrier, f_(L) is the Leeson frequency(equal to the loaded resonator half width half maximum), f_(c) is theobserved noise corner of the sustaining amplifier under operatingconditions, P_(s) is the input power to the sustaining amplifier, T isthe temperature (assumed for simplicity to be the same for both theamplifier and the resonator), k_(B) is Boltzmann's constant, and F isthe sustaining amplifier wideband noise factor. Although phase noise ismost commonly discussed as a power spectral density,

(f_(m)), the square root of the power spectral density,

^(1/2)(f_(m)), is often more relevant for the discussion here. Here,

(f_(m)) is called the phase-noise power spectral density and

^(1/2)(f_(m)) is called the phase-noise spectral density.

Leeson's equation results in an effective gain of noise at frequenciesbelow the Leeson frequency f_(L). Thus, the oscillator architecture issusceptible to additive phase noise sources inside the oscillation loopat frequencies below the Leeson frequency, so care should be taken inparticular to reduce or minimize the noise contribution of thesustaining amplifier (the only active component inside the loop), whichintroduces its flicker noise and wideband noise figure into the system.In contrast, the noise requirements on components outside or after theoscillator magnetometer loop (e.g., buffer amplifiers, mixers,digitizers) may be relatively relaxed. The limited number of criticalcomponents is advantageous for compactness and simplicity of design.

Ferrimagnetic Resonance

In a ferrimagnetic material, strong coupling between nearby electronicspins results in collective spin behavior, including resonances betweencollective spin states. This strong spin-spin coupling can also resultin exchange-narrowing of the ferrimagnetic resonances, thereby allowingsub-MHz transition linewidths to be observed in materials with unpairedspin densities of about 10²² cm⁻³. From Leeson's equation (Eqn. (1)), anarrower resonance results in a lower value of f_(L) and therefore isexpected to result in better phase noise performance of the oscillatormagnetometer. The magnetic material with the narrowest knownferromagnetic resonance linewidth and lowest known spin-wave damping isyttrium iron garnet (YIG), a synthetic, insulating crystal ferrimagnetwith chemical composition Y₃Fe₅O₁₂. Other attractive aspects of YIG arelow acoustic damping (less than that of quartz) and well-developedgrowth processes which yield samples of very high crystal quality.Consequently, YIG is the platform of choice in cavity spintronicsresearch and has found use in magnon-cavity coupling experiments,magneto-acoustic coupling studies, and hybrid quantum circuits, as wellas in axion searches.

In SI units, Kittel's formula for the uniform precessional mode offerromagnetic resonance is ω_(y)=√{square root over([γB_(z)+(N_(y)−N_(z))γμ₀M_(z)][γB_(z)+(N_(x)−N_(z))γμ₀M_(z)])}, where γis the electron gyromagnetic ratio, B=B_(z){circumflex over (z)} is theapplied magnetic field (which we take to define the {circumflex over(z)} axis of the system), M_(z) is the magnetization (assumed to besaturated), and N_(x), N_(y), and N_(z) are the demagnetization factors.For a spherical sample, N_(x)=N_(y)=N_(z)=⅓, and Kittel's formulareduces toω_(y) =γB _(z).  (2)As presented here, Kittel's equations neglect crystal anisotropy,whereby additional Hamiltonian terms arise from electron interactionswith the electric fields of the crystal lattice. These effects, whenpresent, can be treated as perturbative corrections as explained below.

A ferrimagnetic resonance (FMR) can be employed as a frequencydiscriminator, passing signals with frequencies near ω_(y) whilerejecting all others. Consider a geometry consisting of two orthogonalsemicircular coupling loops with a small ferrimagnetic sphere centeredat the intersection of the loop axes, as shown in FIG. 1A, and describedbelow. In the presence of an externally applied DC magnetic bias field(B=B₀{circumflex over (z)}), magnetic domains within the sample alignalong {circumflex over (z)}, magnetizing the sphere. Microwave (MW)drive fields with angular frequency ω_(d)≈ω_(y), applied to the inputcoupling loop, induce precession of the sphere's magnetization about the{circumflex over (z)} axis. The precessing magnetization theninductively couples to the output loop (with a π/2 phase shift relativeto the input loop). For this construction, the transmission scatteringparameter S₂₁ obeys

$\begin{matrix}{S_{21} = {\frac{\sqrt{\kappa_{1}\kappa_{2}}}{{i\left( {\omega_{d} - \omega_{y}} \right)} + \frac{\kappa_{0} + \kappa_{1} + \kappa_{2}}{2}}{e^{{- i}\frac{\pi}{2}}.}}} & (3)\end{matrix}$where κ₀, κ₁, and κ₂ are the unloaded FMR linewidth, input couplingrate, and output coupling rate, respectively, in angular frequencyunits. The |S₂₁| transmission exhibits a Lorentzian line shape with amaximum at the FMR frequency ω_(y) and a loaded full-width half-maximum(FWHM) linewidth κ_(L) ≡κ₀+κ₁+κ₂.

Magnetometer performance depends upon localizing the ferrimagneticresonance with precision, accuracy, and speed. The precision with whichthe FMR resonance can be localized (and therefore the ambient magneticfield determined) depends on the intrinsic linewidth of the FMRresonance κ₀. Single-crystal YIG exhibits the lowest linewidth of anyknown ferromagnetic or ferrimagnetic material, with highly polished YIGspheres exhibiting measured linewidths of 2π×560 kHz or below. Thematerial used for the measurements disclosed here exhibits a FWHMlinewidth of about 2π×910 kHz 0.03 mT) at ω_(y)≈2π×5 GHz.

Minimal values of κ₀ arise under uniform orientation of the magneticdomains within the YIG crystal. This is achieved by applying an externalbias magnetic field with sufficient strength to saturate themagnetization. For pure YIG, the saturation magnetization is reachedusing a bias field B₀≈0.178 T. Operation in the saturated magnetizationregime is important to ensure the ferrimagnetic resonance displays aconstant response to changes in the externally applied magnetic field,namely dω_(y)/dB=γ.

Ferromagnetic and ferrimagnetic materials provide much higher unpairedelectron spin density relative to paramagnetic or vapor cell atomicsystems, which translates to small sensing volumes and thus tolerance toenvironmental magnetic field gradients.

In crystallographically perfect YIG, five of every twenty lattice sites(equivalent to one unit formula Y₃Fe₅O₁₂) are populated by trivalentiron (Fe³⁺, electronic spin S=5/2), which occupy three tetrahedrallattice sites and two octahedral lattice sites. Strong superexchangeinteractions (mediated by oxygen ions between the iron ions) align thethree tetrahedral Fe³⁺ antiparallel to the two octahedral lattice Fe³⁺in the absence of thermal excitation. For a single magnetic domain atabsolute zero temperature, YIG exhibits a net magnetic moment equal tothat of one Fe³⁺ atom per every 20 lattice atoms, resulting in apolarized electron spin density of 2.1×10²²/cm³. Magnetization at roomtemperature retains 72% of the maximal magnetization, equal to apolarized electronic spin density of 1.5×10²²/cm³. For comparison,typical paramagnetic spin systems exhibit spin densities within a feworders of magnitude of 10¹⁷/cm³, while alkali vapor cells operate in thevicinity of 10¹³/cm³.

The high spin density and strong coupling between spins, which preventsdeleterious broadening, allows sensitivities of roughly 100 fT/√{squareroot over (Hz)} or better to be achieved using crystal volumes of ≲1mm³. Magnetic gradients can compromise sensitivity when the gradientbroadening becomes comparable to the intrinsic linewidth. Assuming anintrinsic linewidth of κ₀=2π×1 MHz results in a gradient tolerance ofapproximately 0.4 mT/cm before substantial degradation of sensorperformance is expected. This gradient tolerance compares favorably tothe roughly 30 nT/cm gradient tolerance characteristic of alkali vapormagnetometers (which exhibit sample volume length scales about ten timeslarger and intrinsic linewidths about one-thousand times smaller thanthe ferrimagnetic oscillator magnetometers disclosed here).

Changes in the external DC magnetic field at the magnetometer alter theFMR frequency ω_(y), and therefore the oscillator output frequency. TheFMR frequency also responds to AC magnetic fields, and the process bywhich AC fields alter the magnetometer output waveform is disclosedbelow. Operationally, AC magnetic fields are encoded via frequencymodulation into the oscillator's output waveform. For example, asingle-frequency AC magnetic field with root-mean-square (rms) amplitudeB_(sen) ^(rms) and angular frequency ω_(m) produces two sidebands at±ω_(m) relative to the oscillator carrier frequency. These two sidebandseach exhibit a carrier-normalized amplitude of

$\begin{matrix}{s = {\frac{\gamma B_{sen}^{rms}}{\sqrt{2}\omega_{m}}.}} & (4)\end{matrix}$

The oscillator magnetometer's sensitivity can then be determined fromthe sideband amplitude along with the measured phase noise

^(1/2) (f_(m)), which represents the background against which thesidebands are discerned. The expected sensitivity is

$\begin{matrix}{{\eta\left( f_{m} \right)} = {\frac{f_{m}}{\gamma/\left( {2\pi} \right)}{{\mathcal{L}^{\frac{1}{2}}\left( f_{m} \right)}.}}} & (5)\end{matrix}$

We note a surprising and striking feature of the oscillator magnetometerarchitecture: assuming the oscillator phase noise is well-described byLeeson's equation (Eqn. (1)), the signal s∝1/ω_(m)=1/(2πf_(m)) and thephase noise are expected to exhibit nearly identical scaling within arange of frequencies between the amplifier noise corner f_(c) and theLeeson frequency f_(L). Thus, the sensitivity of the device versusfrequency f_(m) is expected to be approximately flat forf_(c)<f_(m)<f_(L).

Transmission-Geometry Ferrimagnetic Oscillator Magnetometer

FIG. 1A shows a ferrimagnetic oscillator magnetometer 100 that can beused to measure alternating current (AC) or time-varying magneticfields. This magnetometer 100 uses a transmission geometry with fourcomponents connected in serial loop: a ferrimagnetic resonancetransmission filter 110 (which passes signals near the ferrimagneticresonance ω_(y) and rejects signals at other frequencies), a sustainingamplifier 120 (to provide the requisite gain), a directional coupler 130(to sample the oscillator waveform for device output), and an optionalphase shifter 122 (to ensure that the Barkhausen stability criterion issatisfied).

As explained in greater detail, the ferrimagnetic resonance transmissionfilter 110 has resonance whose center frequency shifts in response tothe magnitude and direction of an external magnetic field. Thisferrimagnetic resonance transmission filter 110 acts as a magnetic fieldsensing element and can be implemented as a 1-mm-diameter, highlypolished YIG (Y₃Fe₅O₁₂) sphere, which acts as a transmission cavity.Other suitable ferrimagnetic materials for the filter 110 include butare not limited to Gallium-doped YIG (Ga³⁺:Y₃Fe₅O₁₂), Aluminum-doped YIG(Al³⁺:Y₃Fe₅O₁₂), Lithium Ferrite (Li_(0.5)Fe_(2.5)O₄), or CalciumVanadium Bismuth Iron Garnet (Ca₂VBiFe₄O₁₂). In this example, the YIGsphere is mechanically supported between the dimpled end faces of two3-mm-diameter, 10-mm-long rods (not shown) made of sapphire, beryllia(BeO), or another electrically insulating material with high thermalconductivity.

As shown in FIG. 1A, a semicircular input coupling loop 112 a andsemicircular output coupling loop 112 b (e.g., each of 3.5±0.5 mmradius) provide inductive coupling to and from the YIG sphere 110. Thecoupling loops 112 are mounted orthogonal to each other so that S₂₁transmission can only occur through the gyrator action of theferrimagnetic resonance, with transmission far from resonance suppressedto negligible levels. In practice, the coupling loops 112 do not have tobe exactly orthogonal; minor twisting and positional variation of thecoupling loops 112 should not produce any problematic off-resonantcoupling.

The input coupling rate κ₁ and output coupling rate κ₂ between thefilter 110 and the coupling loops 112 can be adjusted by changing thediameters of the coupling loops, while the intrinsic linewidth κ₀ of thefilter 110 is essentially a fixed property of the device (depending onthe quality of the YIG, the uniformity of the bias magnetic field,etc.). The values of κ₀, κ₁ and κ₂ are determined by simultaneouslymeasuring the S-parameters S₁₁ and S₂₁ on the ferrimagnetic resonancetransmission filter 110. In this example, the total loaded angularlinewidth is κ_(L) ≡κ₀+κ₁+κ₂=2π×1.60 MHz, equivalent to a loaded qualityfactor Q_(L)=3125, so that the Leeson frequency is

${f_{L} \equiv {\frac{1}{2}\frac{\kappa_{L}}{2\pi}}} = {800\mspace{14mu}{{kHz}.}}$For this magnetometer, κ₀=2π×910 kHz, κ₁=2π×370 kHz, and κ₂=2π×320 kHz.

Two cylindrical permanent magnets 114 positioned symmetrically along theaxis {circumflex over (z)} relative to the ferrimagnetic sphere 110apply a uniform bias magnetic field B₀=B₀{circumflex over (z)} as shownin FIG. 1A. The amplitude of the uniform bias magnetic field should behigh enough (e.g., B₀≈0.178 T) to ensure that the sphere's magnetizationis saturated, so that the filter response is governed by ω_(y)(t)=γB(t).Ceramic (strontium ferrite) bias magnets 114 may be employed to reduceor minimize shielding of alternating current (AC) magnetic fields to besensed. The YIG sphere is aligned so that its <111> easy axis lies along{circumflex over (z)}, i.e., parallel to B₀. With this bias magneticfield, the oscillation frequency is about 2π×5 GHz.

The YIG sphere's precessing magnetization continuously induces a voltageon the output coupling loop 112 b, with a frequency equal to themagnetization precession frequency. This 1\4 W voltage signal thenpasses through the 10 dB directional coupler 130. The through port ofthe directional coupler 130 connects to the sustaining amplifier 120followed by the optional (mechanically adjustable) phase shifter 122 asshown in FIG. 1A. For oscillator frequencies at X-band and above, a SiGetransistor or InGaP transistor can be used as the sustaining amplifier120. After the phase shifter 122, the MW signal is directed to the inputcoupling loop 112 a as indicated by the dashed arrows in FIG. 1A, whichinductively couples the MWs back to the YIG's precessing magnetization,closing the oscillator loop. The phase shifter is adjusted to reduce orminimize the device phase noise, which is measured in real time.

The ferrimagnetic crystal 110 hosts an ensemble of entangled electronicspins, which together have a transmission resonance whose centerfrequency shifts in response to an applied magnetic field (e.g., analternating current (AC) magnetic field). The analog microwave signalpropagating through the loop formed by the ferrimagnetic crystal 110 andsustaining amplifier 120 probes this resonance. Changes in the magnitudeor direction of the applied magnetic field shift the resonance of theensemble of entangled electronic spins. The shift in the resonance inturn modulates the analog microwave signal, producing upper and lowersidebands whose amplitudes are proportional to the magnitude of theapplied magnetic field. The coupler 130 has a tap port that diverts aportion (e.g., 1%, 5%, or 10%) of this sideband-modulated analogmicrowave signal to the digitizer 140, which generates a digital versionof the signal that can be used sense the sidebands and hence the (changein) direction and/or magnitude, frequency, or phase of the appliedmagnetic field.

The ferrimagnetic oscillator magnetometer 100 is like other oscillatorsin that it includes a frequency-selective element (the ferrimagneticcrystal 110) and a gain mechanism (the sustaining amplifier 120). Theferrimagnetic resonance has a center frequency and a quality factor Q.It is causal (like everything in the real world), so it has a pathlength (and a delay). As a result, the ferrimagnetic resonance acts justlike a RF/MW filter cavity. Microwaves input into the input couplingloop 112 a couple to the output coupling loop 112 b by coupling throughthe spins precessing at the ferrimagnetic resonance. Signals atfrequencies near the ferrimagnetic resonance are passed and frequenciesaway from the ferrimagnetic resonance are strongly attenuated. Thus, theferrimagnetic element 110 acts as a frequency filter by passingfrequencies within a linewidth of the ferrimagnetic resonance andrejecting other frequencies. An optional bandpass filter (not shown) maybe used to suppress spurious frequencies. The sustaining amplifier 120has gain near the ferrimagnetic resonance frequency and makes up for theround-trip loss of the oscillator.

Frequency selectivity is enforced both by the amplitude of transitionthrough the ferrimagnetic resonance and by setting the round-trip phaseto be an integer multiple of 2n. For the ferrimagnetic oscillatormagnetometer 100, most of the path length comes from the high-QRF/microwave cavity formed by the YIG sphere 110. (At a frequency of 5GHz, the wavelength in air is 6 cm, so for a loaded quality factorQ=5000, the effective path length will be approximately 300 meters.) Ifthe optional bandpass filter passes frequencies within 100 MHz or so ofthe ferrimagnetic resonance, the oscillator should automaticallyoscillate at the ferrimagnetic resonance without any external microwavesor laser.

In one example, under typical operating conditions, the input power tothe sustaining amplifier 120 is P_(s)=−3 dBm. The sustaining amplifier120 has a measured gain of 11.4 dB at P_(s)=−3 dBm, so that, afteraccounting for roughly 1 dB of additional loss, about 7.4 dBm of MWpower is delivered to the input coupling loop 112 a. The sustainingamplifier 120 is powered by a 12-volt lead-acid battery (not shown) topartially mitigate the additive phase noise induced by amplifier powersupply voltage fluctuations.

The 10 dB directional coupler 130 allows sampling of the oscillator'svoltage waveform; measurement of this waveform allows reconstruction ofthe magnetic field waveform as described below. The output of coupledport of the directional coupler 130 is coupled to a 1×2 switch 132,which has one port coupled directly to a phase-noise analyzer 150 (fordevice diagnostics) and one port coupled to a mixer 136 via a bufferamplifier 134. The phase-noise analyzer 150 can be used to performdiagnostics and device optimization. The mixer 136 is coupled to a localoscillator 138 mixes the signal down to an intermediate frequency,ω_(i), in a range (e.g., MHz) appropriate for a digitizer 140.

FIG. 1B shows an implementation of the digitizer 140 that converts theanalog, frequency-modulated signal from the ferrimagnetic oscillatormagnetometer 100 into in-phase and quadrature intermediate-frequencysignals, which are then converted into digital signals. A mixer 142mixes an analog output 101 from a ferrimagnetic oscillator magnetometerwith a local oscillator (LO) from a low-phase-noise reference oscillator148 to produce in-phase and quadrature components that can be digitizedby respective analog-to-digital converters (ADCs) 144 and 146. Thesecomponents can be demodulated as described below.

Equivalent Circuit for YIG, S-Parameters, and Intrinsic Linewidth

FIG. 1C shows an equivalent circuit for the YIG sphere 110 withorthogonal coupling loops 112 (e.g., two loops in planes rotated fromeach other by π/2, the geometry of this device, shown as impedances Z₀).This circuit closely resembles the equivalent circuit for a series RLCcircuit with inductive coupling. One difference is that the gyratoraction of the YIG sphere 110 introduces a direction-dependent (i.e.,non-reciprocal) phase shift: S₂₁ is retarded by π/2 while S₁₂ isadvanced by π/2. The S parameters describing this system are

$\begin{matrix}{S_{11} = {1 - \frac{\kappa_{1}}{{i\left( {\omega_{d} - \omega_{y}} \right)} + {\left( {\kappa_{0} + \kappa_{1} + \kappa_{2}} \right)/2}}}} & (6) \\{S_{21} = {\frac{\sqrt{\kappa_{1}\kappa_{2}}}{{i\left( {\omega_{d} - \omega_{y}} \right)} + {\left( {\kappa_{0} + \kappa_{1} + \kappa_{2}} \right)/2}}e^{{- i}{\pi/2}}}} & (7) \\{S_{12} = {\frac{\sqrt{\kappa_{1}\kappa_{2}}}{{i\left( {\omega_{d} - \omega_{y}} \right)} + {\left( {\kappa_{0} + \kappa_{1} + \kappa_{2}} \right)/2}}e^{i{\pi/2}}}} & (8) \\{S_{22} = {1 - \frac{\kappa_{2}}{{i\left( {\omega_{d} - \omega_{y}} \right)} + {\left( {\kappa_{0} + \kappa_{1} + \kappa_{2}} \right)/2}}}} & (9)\end{matrix}$where ω_(d) is the drive frequency, ω_(y) is the ferrimagnetic resonancefrequency, κ₀=R/L is the intrinsic YIG linewidth, κ₁=Z₀/n₁ ²L is theinput coupling rate, and κ₂=Z₀/n₂ ²L is the output coupling rate. Theparameters ω_(d), ω_(y), κ₀, κ₁, and κ₂ are all in angular units. Theintrinsic linewidth of the ferrimagnetic resonance filter 110 is givenby κ₀=ω_(y)/Q₀, where Q₀ is the unloaded quality factor of the YIGsphere, extracted from measurements performed by sweeping ω_(d) while B₀is fixed. The intrinsic linewidth can also be determined by sweeping thevalue of B₀ for a fixed value of ω_(d), and literature values of YIGlinewidths are often given in magnetic field units rather than frequencyunits. The S-parameter equations above are valid only near resonance,both because of the inclusion of the ideal transformer and because theequations have been symmetrized about the resonance frequency.

We wish to determine the intrinsic linewidth κ₀ of the uniform modeferrimagnetic resonance from S-parameter data measured by a vectornetwork analyzer (VNA) on a two-port YIG transmission filter, as shownin FIG. 1A. From the Eqns. (6) and (7), the maximum and minimum of thepower transmission and reflection are, respectively,

${{S_{11}}_{\min}^{2} = \left( {1 - \frac{\kappa_{1}}{\left( \frac{\kappa_{0} + \kappa_{1} + \kappa_{2}}{2} \right)}} \right)^{2}},{{S_{11}}_{\max}^{2} = {\frac{\kappa_{1}\kappa_{2}}{\left( \frac{\kappa_{0} + \kappa_{1} + \kappa_{2}}{2} \right)^{2}}.}}$

The value of the loaded linewidth, κ_(L)=κ₀+κ₁+κ₂, can be determinedfrom the distance between the points where |S₂₁|² is reduced by 3 dBfrom its peak value. The system can then be solved for κ₀, κ₁, and κ₂,as there are three equations and three unknowns:

${\kappa_{0} = {\frac{\kappa_{L}}{2}\left( {1 + \sqrt{{S_{11}}_{\min}^{2}} - \frac{{S_{11}}_{\max}^{2}}{1 - \sqrt{{S_{11}}_{\max}^{2}}}} \right)}},{\kappa_{1} = {\frac{\kappa_{L}}{2}\left( {1 - \sqrt{{S_{11}}_{\min}^{2}}} \right)}},{\kappa_{2} = {\frac{\kappa_{L}}{2}{\left( \frac{{S_{11}}_{\max}^{2}}{1 - \sqrt{{S_{11}}_{\min}^{2}}} \right).}}}$Ferrimagnetic Oscillator Magnetometer Operation

FIGS. 2A-2C illustrates the principles of operation of the ferrimagneticoscillator magnetometer 100 shown in FIG. 1A. FIG. 2A shows theferrimagnetic YIG sphere 110 subjected to a uniform external magneticfield B from the bias magnet 114 (FIG. 1A). This uniform magnetic fieldcauses the electronic spins of the ferrimagnetic YIG sphere 110 toprecess in phase at the resonance frequency ω_(y). (FIG. 2A shows fourspins, each depicted as an arrow following a corresponding dashedcircular track.) The resonance (and precession) frequency ω_(y) of theuniform ferrimagnetic resonance mode varies linearly with the totalmagnetic field seen by the sphere as shown in FIG. 2B. When theferrimagnetic resonance is employed as a frequency discriminator in aferrimagnetic oscillator magnetometer, the oscillation frequency of theferrimagnetic oscillator magnetometer tracks the ferrimagnetic resonancefrequency. As a result, changes in the amplitude of the magnetic fieldcause the oscillation frequency to change.

FIG. 2C shows how the amplitude of the magnetic field B (t) applied tothe magnetometer is encoded in frequency modulation of the oscillator'soutput waveform. Decreasing the magnetic field amplitude reduces theoscillation frequency. And increasing the magnetic field amplitudeincreases the oscillation frequency. By demodulating the outputwaveform, the original time-domain magnetic field signal B(t) may berecovered.

For the uniform mode of ferrimagnetic resonance in a spherical sample(e.g., the YIG sphere 110 in the magnetometer 100 of FIG. 1A) withsaturated magnetization, the time derivative of the instantaneous phaseϕ(t) obeys

$\begin{matrix}{{\frac{d{\phi(t)}}{dt} = {\gamma{B(t)}}},} & (10)\end{matrix}$

where B(t) is the externally applied magnetic field and γ=g_(e)μ_(B)/ℏ(with the electron g-factor g_(e)≈2, the Bohr magneton μ_(B), and thereduced Planck's constant ℏ, so that γ≈2π×28 GHz/Tesla). (For thisdiscussion, we neglect crystal anisotropy, which introduces higher orderterms into Eqn. (10).)

The precessing magnetization of the sphere described by Eqn. (10)inductively couples to the output coil (e.g., coil 112 b in FIG. 1A),producing a voltage signal which is then amplified by the sustainingamplifier (e.g., amplifier 120 in FIG. 1A) and inductively coupled backto the precessing magnetization; this closed loop produces sustainedself-oscillation. The oscillator output voltage is then v(t)=V₀cos[ϕ(t)] where V₀ corresponds to the oscillator's voltage amplitude.Since the oscillating voltage at any point in the loop has a fixed phaserelative to the magnetization, we do not distinguish between the phaseof the magnetization and the phase of the oscillator's voltage output.

The oscillator phase ϕ(t) is continuous in time and given by

${{\phi(t)} = {{\int_{0}^{t}{\frac{d{\phi(\tau)}}{d\tau}d\tau}} = {\int_{0}^{t}{\gamma{B(\tau)}d\tau}}}},$where ϕ(t=0)≡0. The total magnetic field B(t) seen by the magnetometeris the vector sum of a static field B₀=B₀{circumflex over (z)} (createdby the permanent magnets 114) and the ambient field B_(sen)(t) externalto the magnetometer. For simplicity, assume that B_(sen)(t) lies along{circumflex over (z)} (the case of arbitrary B_(sen)(t) is describedbelow). The value of B₀ is assumed to exhibit only slow temporalvariation (due to thermal drift of the magnets or vibration of themechanical structure holding the magnets) on time scales below thefrequencies of interest, so that B₀ can be treated as constant. Thetotal magnetic field seen by the ferrimagnetic sphere isB(t)=B₀+B_(sen)(t), allowing the oscillator phase to be expressed asϕ(t)=∫₀ ^(t)γ[B ₀ +B _(sen)(τ)]dτ.  (11)

An arbitrary real waveform B_(sen)(t) can be decomposed into its Fourierseries as

${{B_{sen}(t)} = {\frac{a_{0}}{2} + {\sum\limits_{n = 1}^{\infty}{a_{n}{\cos\left( {\omega_{n}t} \right)}}} + {\sum\limits_{n = 1}^{\infty}{b_{n}{\sin\left( {\omega_{n}t} \right)}}}}}.$

For simplicity, assume that B_(sen)(t) consists of a single spectralcomponent such that B_(sen)(t)=√{square root over (2)}B_(sen) ^(rms)[cosω_(m)t], where ω_(m) is the angular frequency of the magnetic field andB_(sen) ^(rms) is the rms field amplitude. With this simplification, theoscillator phase is

$\begin{matrix}{{\phi(t)} = {\int_{0}^{t}{{\gamma\left\lbrack {B_{0} + {\sqrt{2}{B_{sen}^{rms}\left\lbrack {\cos\omega_{m}\tau} \right\rbrack}}} \right\rbrack}d\;\tau}}} \\{= {{\gamma B_{0}t} + {\sqrt{2}{{\frac{\gamma B_{sen}^{rms}}{\omega_{m}}\left\lbrack {\sin\omega_{m}t} \right\rbrack}.}}}}\end{matrix}$The oscillator waveform (e.g., as in the upper trace of FIG. 2C) is then

${v(t)} = {V_{0}{{\cos\left\lbrack {{\gamma B_{0}t} + {\sqrt{2}\frac{\gamma B_{sen}^{rms}}{\omega_{m}}{\sin\left\lbrack {\omega_{m}t} \right\rbrack}}} \right\rbrack}.}}$

This expression for the oscillator waveform can be transformed using aBessel function identity and subsequently Taylor expanded (with γB_(sen)^(rms)<<ω_(m)) to give

${{v(t)} = {V_{0}\left\lbrack {{\cos\left\lbrack {\gamma B_{0}t} \right\rbrack} + {\frac{\gamma B_{sen}^{rms}}{\sqrt{2}\omega_{m}}{\cos\left\lbrack {\left( {\omega_{c} + \omega_{m}} \right)t} \right\rbrack}} - {\frac{\gamma B_{sen}^{rms}}{\sqrt{2}\omega_{m}}{\cos\left\lbrack {\left( {\omega_{c} - \omega_{m}} \right)t} \right\rbrack}} + {H.O.}} \right\rbrack}},$where “H.O.” represents higher-order terms in the Taylor expansion. Forexternal fields satisfying γB_(sen) ^(rms)<<ω_(m), the B-field frequencymodulation results in two antisymmetric sidebands at ±ω_(m), each withamplitude γB_(sen) ^(rms)/(√{square root over (2)}ω_(m)). For example, a1 pT RMS magnetic field at 100 kHz produces two sidebands each withpower −134 dBc.

The oscillator's instantaneous phase ϕ(t) is governed by Eqn. (7), wherethe bias field from the bias magnet(s) is large enough to saturate themagnetization of the ferrimagnetic material. Differentiating theoscillator's instantaneous phase yields

$\frac{d{\phi(t)}}{dt} = {{\gamma\left\lbrack {B_{0} + {B_{sen}(t)}} \right\rbrack}.}$

The time domain magnetic field waveform B_(sen)(t) is then determined bycalculating

${B_{sen}(t)} = {\frac{\frac{d{\phi(t)}}{dt}}{\gamma} - {B_{0}.}}$

As a practical matter, the demodulation process can be facilitated byapplying the Hilbert transform to the (real-valued) voltage waveform ofthe oscillator, producing a complex signal that allows the instantaneousphase ϕ(t) to be determined in a simple manner. The phase is thenunwrapped if appropriate so that it is continuous and free from 2πjumps, and finally dϕ(t)/dt is calculated numerically.

The instantaneous phase of the oscillator encodes the value of themagnetic field. The Hilbert transform allows the instantaneous phaseϕ(t) to be determined in isolation from variations in the instantaneousamplitude. There are two conditions on the Hilbert transform to achievethis objective: First, the additive phase noise φ(t) should be small,i.e., |φ(t)|<<1; and second, the additive phase noise φ(t) and additiveamplitude noise α(t) should vary slowly compared to the intermediatefrequency out of the mixer ω_(i) (that is, they should have negligiblefrequency components above ω_(i)). The first and second conditions holdfor the ferrimagnetic oscillator magnetometers disclosed here.

The output of the mixer is digitized and may be written as a real-valuedwaveform,v(t)=V ₀[1+α(t)]cos[ω_(i) t+φ(t)].  (12)As |φ(t)|<<1 (the first condition), trigonometric identities and theapproximations cos φ(t)≈1 and sin φ(t)≈φ(t) allow Eqn. (12) to berewritten asv(t)≈V ₀[1+α(t)][cos[ω_(i) t]−φ(t)sin[ω_(i) t]].

From Bedrosian's theorem, the second condition (that α(t) and φ(t) varyslowly compared to ω_(i)) allows the Hilbert transform of v(t) to becalculated by transforming only the high frequency componentscos[ω_(i)t] and sin[ω_(i)t]. Denoting the Hilbert transform of v(t) as{circumflex over (v)}(t) yields{circumflex over (v)}(t)≈V ₀[1+α(t)][sin[ω_(i) t]+φ(t)cos[ω_(i) t]].

Using small angle approximations (the first condition) and trigonometricidentities gives the resulting analytic signal:v(t)+i{circumflex over (v)}(t)≈V ₀[1+α(t)]e ^(i(ω) ^(i) ^(t+φ(t))).

The instantaneous phase of the mixed-down signal ω_(i)t+φ(t) can bedetermined by taking the argument of the above analytic signal. As thequantity [1+α(t)] is common to both the real and imaginary components,the additive amplitude noise α(t) is thereby isolated from theinstantaneous phase. This approach should be compared to the real-valuedwaveform of Eqn. (12) where there is no direct way to isolate theinstantaneous phase from additive amplitude noise.

Note that φ(t) is real, so its double-sided power spectrum is symmetricabout zero frequency. Therefore, in the frequency domain picture, it maynot be possible to gain any sensitivity by processing both the positiveand negative frequency sidebands, as their information is redundant.

FIG. 3A illustrates a process for determining the AC magnetic fieldamplitude from the instantaneous phase of the real voltage waveformoscillating in a ferrimagnetic oscillator magnetometer according to thesteps laid out above. (The plot below each box in FIG. 3A shows asimulated representative waveform corresponding to that step of theprocess.) Applying a magnetic field to the ferrimagnetic oscillatormagnetometer (302) produces a real oscillator voltage waveform whoseinstantaneous phase varies with the magnetic field amplitude asdescribed above. This voltage waveform can be digitized and measured(304), then Hilbert-transformed to reconstruct the complex waveform(306), which can be used to check the phase noise (307) and to determinethe oscillator phase angle as a function of time (308). Removing thephase evolution caused by the DC (bias) magnetic field (310) anddifferentiating the phase angle in time and scaling the amplitude of theresult (312) yields the measured AC magnetic field amplitude. FIG. 3Bshows that this recovered magnetic field waveform is consistent withtrue magnetic field waveform to within sensitivity/noise of the device.If desired, the recovered magnetic field waveform can be Fouriertransformed to yield the power spectral density of the AC magnetic field(314).

Experimental Measurements with a Transmission-Geometry FerrimagneticOscillator

The sensitivity of a magnetometer may be determined from themagnetometer response to a known applied field along with the measurednoise. Because AC magnetic fields are encoded by the oscillatormagnetometer as frequency modulation of its roughly 5 GHz outputwaveform, the measured phase noise may limit the magnetic sensitivity ofthe device.

FIG. 4A shows measured single-sideband phase-noise power spectraldensity

(f_(m)) for the ferrimagnetic oscillator magnetometer of FIG. 1A. Thedevice can operate with a phase noise of −122.5 dBc/Hz at 10 kHz offsetand −148.5 dBc/Hz at 100 kHz offset. Fitting Eqn. (1) to thesingle-sided phase noise to the phase-noise data gives f_(L)=800 kHz,F=2, and f_(c)=80 kHz (with the measured P_(s)=−3 dBm during operation).The fit value of f_(L) is consistent with the value expected from themeasured loaded linewidth of the ferrimagnetic resonance transmissionfilter element. The noise factor F=2 is consistent with themanufacturer-specified wideband noise figure of 3 dB.

FIG. 4B shows the predicted and measured frequency response of theferrimagnetic oscillator magnetometer of FIG. 1A to a 150 nT rms ACmagnetic field applied along the 2 axis. The predicted response iscalculated according to Eqn. (4). A sinusoidal magnetic field with rmsamplitude B_(sen) ^(rms)=150 nT was applied to the sensor, the angularfrequency ω_(m) was varied, and the carrier-normalized amplitude of theresulting sidebands is recorded. The data are in excellent agreementwith Eqn. (4), as shown in FIG. 4B.

FIG. 4C shows the device sensitivity spectrum as derived from the deviceresponse in FIG. 4B. As noted above, the sensitivity should beapproximately flat in the region between the amplifier noise corner atf_(c)≈80 kHz and the Leeson frequency f_(L)≈800 kHz. The measured dataare consistent with this expectation; for AC signals of known phase, theminimum sensitivity is 140 fT/√{square root over (Hz)} and thesensitivity is below 300 fT/√{square root over (Hz)} over the band from10 kHz to 1 MHz. For AC signals of unknown phase, the sensitivity may bedegraded by a factor of √{square root over (2)} compared to the spectrumshown in FIG. 4C.

The ferrimagnetic oscillator magnetometer disclosed and demonstratedhere provides the best sensitivity achieved to date for a solid-statequantum magnetometer, with sensitivity surpassed only by cryogenic SQUIDmagnetometers and vacuum-based vapor cell magnetometers. Improved devicesensitivity may come either from increasing the signal for a givenmagnetic field or from decreasing the phase noise. Increased signal maybe realized by employing strong cavity coupling schemes which, undercertain conditions, could allow the device frequency response versusmagnetic field to be increased beyond γ=2π×28 GHz/T (see Eqn. (2)).

The device phase noise can be reduced as well. For example, thesustaining power P_(s) can be increased, though this approach may notreduce phase noise much, if at all, at frequencies below f_(c) (andf_(c) itself may increase with larger values of P_(s)). The sustainingpower P_(s) may be limited by instabilities caused by non-linearcoupling of the uniform precession mode to undesired spin wave modes.

Another approach is to reduce the sustaining amplifier's contribution tophase noise. The amplifier's principal noise contributions are flickernoise below f_(c) and wideband noise described by its noise figure.Amplifier-induced noise can be partially mitigated usingoscillator-narrowing techniques such as Pound-Drever-Hall locking,carrier suppression interferometric methods, careful design, or othermethods. However, in the absence of technical noise sources (such asamplifier noise), an idealized oscillator magnetometer should exhibitthe same sensitivity as an idealized transmission interferometermagnetometer. That is, oscillator-narrowing techniques may not reducephase noise to the thermal noise limit (−177 dBm/Hz at room temperature)expected in the absence of Leeson gain (and in the absence of technicalnoise sources). While lowering the Leeson frequency f_(L) should improvephase noise performance, the thermal noise gain introduced by the Leesoneffect appears to be fundamental to the oscillator architecture.

Finally, the encoding of the signal in frequency rather than amplitudemay allow additional techniques developed for precision timekeeping tobe harnessed for improved performance. Indeed, the mature state of clocktechnology is already taken advantage of in the device to some extent.For example, down conversion can be done by mixing the oscillatormagnetometer output with the reference signal provided by an oscillatorof superior phase noise.

In conclusion, the magnetometer design reported here offers a uniquecombination of state-of-the-art sensitivity (with realistic prospectsfor improvement), high dynamic range, compactness, and low powerrequirements. These advantages could drive widespread adoption ofsimilar quantum sensing devices soon. The oscillator architecture can beadapted to simplify high-performance ensemble sensing with a range ofquantum materials and in a variety of sensing modalities, such assensing of electric fields, temperature, or pressure.

Ferrimagnetic Oscillator Magnetometer with Pound-Galani Architectures

FIG. 5A shows a ferrimagnetic oscillator magnetometer 200 in aPound-Galani oscillator architecture with feedback. Like themagnetometer 100 in FIG. 1A, this magnetometer 200 includes aferrimagnetic crystal 210 arranged in a transmission loop with asustaining amplifier 220 and a directional coupler 230. It also includesa feedback loop that generates a Pound-Drever-Hall (PDH) error signal261. This error signal 261 allows the magnetometer 200 to follow thecenter frequency of ferrimagnetic resonance more closely. This cansubstantially improve sensitivity, e.g., by a factor of up to 100 atcertain frequencies.

The magnetometer 200 includes coupling loops 212 that couple a microwave(MW) signal 211 into and out of ferrimagnetic crystal 210. Thisferrimagnetic crystal 210 is subject to a bias magnetic field from abias magnet 214 and has a ferrimagnetic resonance whose center frequencyvaries with the applied magnetic field. The directional coupler 230 tapsa portion of the microwave signal 211 out of the transmission loop as aMW output 231, which is digitized by a digitizer 240 for demodulation asdescribed below. An optional bandpass filter 224 (e.g., with a passbandthat is about 100 MHz wide) between the through port of the directionalcoupler 230 and the input to the sustaining amplifier 220 suppressesnoise and prevents the magnetometer 200 from oscillating at frequenciesnot related to the desired ferrimagnetic resonance frequency.

The feedback loop is implemented with a modulation source 270, such asan RF synthesizer, that generates a continuous-wave (CW) microwavesignal. This signal drives a phase modulator 272 that is coupled in themain transmission loop between the output of the sustaining amplifier220 and right before the first port of a three-port circulator 260. Thesecond port of the three-port circulator 260 is coupled to the inputcoupling loop 212 to the ferrimagnetic crystal 210 and the third port ofthe three-port circulator 260 is coupled to an amplitude detector 262,which is shown in FIG. 5A as a diode. The output of the amplitudedetector 262 feeds the RF port of a mixer 264, which mixes the outputwith another copy of the CW microwave signal from the modulation source270. A low-pass filter 266 filters the down-converted IF output toproduce an error signal 261. A proportional-integral-derivative (PID)controller 268 uses this error signal 261 to drive another phasemodulator 274 to servo the error to zero. This feedback causes theoscillator's output frequency (i.e., the frequency of the MW output 231)to follow the center frequency of the ferrimagnetic resonance moretightly.

FIG. 5B illustrates how the magnetic field amplitude is encoded in thePDH error signal 261. When the oscillation frequency exactly coincideswith the ferrimagnetic resonance frequency, the PDH error signal 261 iszero. When the oscillation frequency is different from the ferrimagneticresonance, the error signal 261 is generated with a value proportionalto the difference between the oscillation frequency and theferrimagnetic resonance frequency.

Reflection-Geometry Ferrimagnetic Oscillator Magnetometer

FIGS. 6A and 6B illustrate a reflection-geometry ferrimagneticoscillator magnetometer 300 and its equivalent circuit, respectively. Inthis geometry, a ferrimagnetic crystal 310 (here, a YIG sphere) issubject to a bias magnetic field from a permanent magnet 314. Theferrimagnetic crystal 310 is also inductively coupled to the emitter eof a bipolar junction transistor (BJT) 320 via an inductive couplingloop 312. The ferrimagnetic crystal 310 can be modeled as a resistor,capacitor, and inductor coupled in parallel to the inductive couplingloop 312 as shown in FIG. 6B. The collector c of the BJT 320 is coupledto a digitizer 340 via an impedance-matching element or network 330. Asin the transmission geometry, applying an external magnetic field to theferrimagnetic crystal 310 shifts the resonance of an ensemble ofentangled electronic spins in the ferrimagnetic crystal. This shiftmodulates sidebands onto a microwave signal reflected by theferrimagnetic crystal 310 to the BJT 320 and coupled by the BJT 320 tothe digitizer 340 for digitization and detection.

Measurements with a Reflection-Geometry Ferrimagnetic OscillatorMagnetometer

FIGS. 7A-7C illustrate a ferrimagnetic oscillator magnetometer device400 made with the reflection geometry illustrated in FIGS. 6A and 6B.FIG. 7A shows an exploded view of the magnetometer 400, which includes aferrimagnetic oscillator 402 and circuit board 404. In this case, theferrimagnetic crystal in the ferrimagnetic oscillator 402 is aferrimagnetic YIG sphere 410 with a diameter of about 400 μm mounted onthe end of a beryllia rod 416 near a gold coupling wire. The couplinggold wire couples the ferromagnetic sphere 410 to the emitter of a BJT420, which is mounted on the circuit board 404 and contained within acylindrical housing 406. A disk-shaped end cap 408 at one end of thecylindrical housing 406 supports the circuit board 404 and electricalconnections to the digitizer. A 25 mm long NdFeB magnet 414 is locatedwithin the cylindrical housing 406 in front of the YIG sphere. Themagnet 414 has a skin depth of approximately 1.7 mm at 100 kHz andapplies a bias magnetic field to the YIG sphere.

The ferrimagnetic oscillator magnetometer 400 draws 300 mW of power (notincluding the digitizer's power consumption) and occupies a volume of 86cm³ (not including the digitizer). It can sense AC magnetic fields witha sensitivity of 427 fT√Hz at 100 kHz. It was used to measure a 214 pTroot-mean-square (RMS) magnetic field at 100 kHz generated by a coil 61inches away with a 25.5 mm radius and 30 turns. The RMS current throughthe coil was 65 mA as measured by an RMS current meter. The coil isoriented to create a magnetic field at the YIG sphere 410 along thelongitudinal axis of the cylinder, or sensor axis (the most favorabledirection).

FIG. 7D is a plot of the phase and amplitude noise of the magnetometerin FIGS. 7A-7C. The magnetometer's frequency output is fed into a phasenoise analyzer. The analyzer uses the cross-correlation method toevaluate the phase noise of the device under test (here, theferrimagnetic oscillator magnetometer). As the principal noise source inthe ferrimagnetic oscillator magnetometer is phase noise, measurement ofdevice phase noise allows a largely complete characterization of thedevice performance. With the known 214 pT RMS magnetic field applied tothe ferrimagnetic oscillator magnetometer, the device measures a signalof −79 dBc at 100 kHz offset from the carrier (data not shown). As shownin FIG. 7D, the residual phase noise of the device is −133 dBc at 100kHz. Therefore, the power SNR is 54 dB. The magnetometer noise is then214 pT/√{square root over (Hz)} reduced by 10^(54/20)=501), whichresults in a measured sensitivity of η=427 fT/√{square root over (Hz)}for fields at 100 kHz. This sensitivity is roughly in line with thesensitivity estimated from the phase noise and the expected signal fromthe equations above.

For the YIG oscillator in FIGS. 7A-7C, the value of γ is fixed at 2π×2.8MHz/Gauss (the electron gyromagnetic ratio). This implies that thatreducing the YIG oscillator's phase noise should improve the YIGoscillator's performance. For example, increasing the quality factorfrom about 500 (the estimated quality factor) to a theoretical (andpractical) limit of about 10,000. Likewise, the sustaining amplifier'snoise corner can be reduced from 200 kHz to 1 kHz or lower. An optimizedfree-running oscillator may operate with a phase noise of about 30fT/√{square root over (Hz)}. The phase noise can also be substantiallyreduced (e.g., by up to about 50 dB) by employing feedback (e.g., in aPound-Drever-Hall arrangement as in FIG. 5A) instead of a free-runningoscillator.

Coherent Averaging of Frequency-Encoded Magnetic Field Measurements

For signals encoded in an amplitude (e.g., a DC voltage level) the RMSuncertainty of the voltage amplitude after N independent measurementslimited by an additive noise source (e.g., Johnson noise, digitizer readnoise, etc.) varies as δV/√{square root over (N)}, where δV is the RMSadditive voltage noise on a single measurement. Reducing the measurementuncertainty by a factor of ten implies averaging for one hundred timeslonger. Unfortunately, averaging like this is not an effective way tocombat additive noise sources for a device with a 100% duty cyclebecause the device is always on so its time-per-measurement cannot beincreased.

For a magnetic field measurement encoded in the oscillation frequency ofa ferrimagnetic oscillator magnetometer, however, the scaling ismarkedly different. The frequency uncertainty δf scales approximatelyproportional to δV/(VN^(3/2)), where N is the number of samples and V isthe amplitude of the sinusoidal signal. Qualitatively, this scaling canbe understood as follows: The accumulated signal in each frequency binincreases linearly with time (and therefore number of samples), whilethe noise bandwidth in each frequency bin decreases as 1/T, where T isthe total sampling time, resulting in an RMS noise per bin proportionalto 1/√{square root over (T)}. Together, these two effects result in theobserved scaling of frequency uncertainty as 1/T^(3/2). As T isproportional to N, the uncertainty also scales as 1/N^(3/2) where N isthe number of samples. FIG. 8 shows experimental data confirming thisscaling for a ferrimagnetic oscillator magnetometer.

Generally, the digitizer signal-to-noise ratio (SNR) per sample does notvary terribly quickly as sampling rate is increased. For athermal-noise-limited digitizer, voltage SNR decreases by √{square rootover (2)} for every doubling of the sampling rate (i.e., thermallylimited read noise δV is proportional to √{square root over (F_(s))}where F_(s) is the sampling rate). Combining the 1/N^(3/2) scaling withsamples with the √{square root over (F_(s))} scaling of δV withdigitizer sampling rate F_(s) shows the frequency noise from additivenoise sources then varies as 1/Fs. The upside of this approach, calledcoherent processing, is that if a device is limited by additive or readnoise, increasing the sampling rate can suppress this additive noise (atleast until the digitizer's clock jitter becomes the limiting factor).Thus, because a ferrimagnetic oscillator magnetometer encodes its outputin a frequency, it has a massive noise performance advantage overmagnetometers than encode their outputs in amplitude (voltages) orphase.

Ferrimagnetic Crystal Anisotropy

Due to the Coulomb interaction, the wavefunctions of unpaired electronswithin the lattice of a ferrimagnetic crystal may deviate from those ofan isolated atom. The distorted spatial wavefunctions couple to the spinvia the spin-orbit interaction, breaking the isotropy of the spinHamiltonian. This anisotropy affects how readily the ferrimagneticcrystal magnetizes along a given direction (giving rise to easy and hardmagnetization axes) and introduces a direction-dependent term into theferrimagnetic resonance frequency. Although the general calculation offerrimagnetic resonance frequency for arbitrary applied magnetic fieldis somewhat involved, it is instructive to examine a simpler case wherethe external magnetic field is confined to lie in the {110} plane. Underthese conditions, the uniform mode resonant frequency differs fromω_(y)=γB and is instead given to good approximation by

${\omega_{y} = {\gamma\left\lbrack {B + {\frac{K_{1}}{\mu_{0}M_{s}}\left( {2 + {\frac{15}{2}\sin^{4}\theta} - {10\sin^{2}\theta}} \right)}} \right\rbrack}},$where θ is the angle in the {110} plane between the <100>crystallgographic axis and the externally applied magnetic field, and

$\frac{K_{1}}{\mu_{0}M_{s}} \approx {{- {4.3}}mT}$for YIG.

It follows that the alignment of <111> parallel to B, as disclosed here,should result in a resonant frequency higher than γB by ≈2π×160 MHz,while alignment of the hard axis <100> should result in a resonancelower by about 2π×240 MHz. The dependence of the ferrimagnetic resonancefrequency on κ₁/M_(s) is removed for θ=ArcSin √{square root over((10−2√{square root over (10)})/15)}≈29.67°. As the anisotropiccontribution to the ferrimagnetic resonance frequency is additive,anisotropy-induced frequency shifts should not alter the device responseto AC magnetic fields beyond the changes in θ introduced by componentsof B_(sen) perpendicular to B₀. There may be higher-order anisotropiceffects, but these effects should be negligible for YIG.

Projected Fundamental Sensitivity Limits for a Spin Magnetometer

The spin-projection-limited magnetic sensitivity η_(spl) for aspin-based DC magnetometer is

${\eta_{spl} \approx {\frac{\hslash}{g_{e}\mu_{B}}\frac{1}{\sqrt{NT_{2}^{*}}}}},$where N is the number of total spins and T₂* is the free induction decaytime (i.e., dephasing time). Importantly, Eqn. (13) assumes the N spinsare independent. In YIG, there are 4.22×10²¹ unit formula of Y₃Fe₅O₁₂per cm³, with each unit formula contributing 5 unpaired electrons. For a1 mm diameter YIG sphere at room temperature, the number of unpairedspins is N=8×10¹⁸. For a full-width half-maximum (FWHM) unloadedlinewidth of 2π×560 kHz (T₂*≈570 ns), the spin-projection-limitedmagnetic sensitivity is η_(spl)=2.7 aT√{square root over (s)}.

The relevance of this expression as a measure of the fundamental limitsof a ferrimagnetic magnetometer remains unclear, as the strong couplingof nearby spins in a ferrimagnet violates the assumption of independentspins. Indeed, the extremely low spin-projection limit calculated forYIG highlights that the limits of this type of magnetometer likelyshould be understood quite differently than those of its paramagneticcounterparts. While the coupling present in ferrimagnets may allowentanglement-enhanced sensing schemes which surpass the limit imposed byEqn. (13), other expressions may emerge with further study that produceless optimistic fundamental limits. In practice, the coupling betweenspins may produce limits on the power that can be applied to probe theresonance, as this coupling gives rise to degenerate spin wave modescoupled to the uniform precession resonance.

Errors Introduced by Finite Bias Field Magnitude

Nominally, a ferrimagnetic oscillator magnetometer measures theprojection of the external magnetic field B_(sen) along the direction ofthe bias magnetic field B₀ created by the bias magnet(s) (e.g., biasmagnetics 114 in FIG. 1A), allowing the ferrimagnetic oscillatormagnetometer to operate as a vector magnetometer. However, slight errorsare introduced by components of B_(sen) orthogonal to the bias magneticfield B₀. We analyze the origin and magnitude of this error here.

FIG. 9 shows the total field applied to the sensor can be represented asthe vector sum of the bias field from the permanent magnet, which has anamplitude B₀ along the {circumflex over (z)} direction (i.e.,B₀≡B₀{circumflex over (z)}), and the field to be sensed, B_(sen):B=B₀+B_(sen). (The MW magnetic field applied at the ferrimagneticresonance frequency is assumed to oscillate rapidly compared to thesensing bandwidth, so that the MW field contributes negligibly to theeffective total magnetic field.)

Although the magnetometer in fact measures the scalar field, it producesan effective vector measurement. To see how, consider the external fieldas B_(sen)=B_(sen) ^(∥)+B_(sen) ^(⊥), where B_(sen) ^(∥) and B_(sen)^(⊥) are the external field components parallel and perpendicular to B₀,respectively. The scalar field is then

$\begin{matrix}{{B} = \sqrt{B \cdot B}} \\{= \sqrt{\left( {B_{0} + B_{sens}^{\parallel}} \right)^{2} + \left( B_{sens}^{\bot} \right)^{2}}} \\{= {B_{0}\sqrt{\left( {1 + \frac{B_{sens}^{\parallel}}{B_{0}}} \right)^{2} + \left( \frac{B_{sens}^{\bot}}{B_{0}} \right)^{2}}}} \\{= {B_{0}{\sqrt{1 + {2\frac{B_{sens}^{\parallel}}{B_{0}}} + \left( \frac{B_{sens}^{\parallel}}{B_{0}} \right)^{2} + \left( \frac{B_{sens}^{\bot}}{B_{0}} \right)^{2}}.}}}\end{matrix}$

Taylor expanding the final expression above with B_(sen)<<B₀ yields

${B} \approx {B_{0} + B_{sens}^{\parallel} + {\frac{\left( B_{sen}^{\bot} \right)^{2}}{2B_{0}}.}}$

The third term in the above expansion provides an estimate of the error,

${Error}{\approx {\frac{\left( B_{sen}^{\bot} \right)^{2}}{2B_{0}}.}}$The maximum error occurs when B_(sen) is oriented perpendicular to the zaxis. The error is minimized when B_(sen) is parallel to the z-axis. Fora 0.05 mT external field and B₀=0.178 T, the maximal error is 7 nT.

These errors may be suppressed (e.g., by a factor of 1000 or more) byassembling a full vector magnetometer out of three ferrimagneticoscillator magnetometers oriented in different (e.g., orthogonal)directions. In this configuration, the measured values from each of thethree sensors can be combined to refine the reconstructed magnetic fieldvector, though this procedure may impose restrictions on the allowablemagnetic field gradient (depending on how closely co-located the threesensors can be).

YIG Magnetometer Noise

As detailed above, a single-frequency AC magnetic field applied to thesensor results in frequency modulation of the oscillator's carrier atangular frequency ω_(m). In the frequency domain, this modulationmanifests as two sidebands offset by ±ω_(m) from the oscillator'scarrier frequency, each with carrier-normalized amplitude γB_(sen)^(rms)/(√{square root over (2)}ω_(m)). Magnetic field detection thenreduces to resolving these two sidebands from the oscillator's measuredphase noise. The magnetic sensitivity η(f_(m)) may be written as a ratiobetween the phase noise spectral density

^(1/2)(f_(m)) and the signal due to these FM sidebands, each withcarrier-normalized amplitude γB_(sen) ^(rms)/(√{square root over(2)}ω_(m)). Thus, the sensitivity is

$\begin{matrix}{{\eta\left( f_{m} \right)} = {\frac{f_{m}}{\gamma/\left( {2\pi} \right)} \times {\mathcal{L}^{\frac{1}{2}}\left( f_{m} \right)}}} & (14)\end{matrix}$where

^(1/2)(f_(m)) is the single-sided phase-noise spectral density of theoscillator. With optimal (synchronous) detection, the SNR is improved by√{square root over (2)} over that expected from a single sideband and aB-field of unknown phase; this factor has been applied to Eqn. (14). Forrealistic oscillators, the phase noise is symmetric about the carrier,and thus there is no improvement to be gained by processing both theupper and lower sideband.

It is informative to apply Leeson's model of oscillator phase noise toEqn. (14). Leeson's equation for the single-sided phase noise of anoscillator as a function of the offset frequency f_(m) from the carrieris given by

$\begin{matrix}{{{\mathcal{L}^{\frac{1}{2}}\left( f_{m} \right)} = \sqrt{{{\frac{1}{2}\left\lbrack {\frac{f_{L}^{2}}{f_{m}^{2}} + 1} \right\rbrack}\left\lbrack {\frac{f_{c}}{f_{m}} + 1} \right\rbrack}\left\lbrack \frac{Fk_{B}T}{P_{s}} \right\rbrack}},} & (15)\end{matrix}$where f_(L) ≡(½)(κ_(L)/2π) denotes the Leeson frequency, f_(c) is theobserved noise corner of the sustaining amplifier under operatingconditions, P_(s) is the sustaining power (i.e., the input to thesustaining amplifier), T is the temperature (assumed for simplicity tobe the same for both the amplifier and the resonator), k_(B) isBoltzmann's constant, and F denotes the wideband noise factor of thesustaining amplifier. Here, κ_(L) is an angular frequency FWHM whilef_(L) is a non-angular frequency half-width. Combining Eqns. (14) and(15) yields an expected sensitivity of

$\begin{matrix}{{\eta\left( f_{m} \right)} = {\frac{f_{m}}{\gamma/\left( {2\pi} \right)}{\sqrt{{{\frac{1}{2}\left\lbrack {\frac{f_{L}^{2}}{f_{m}^{2}} + 1} \right\rbrack}\left\lbrack {\frac{f_{c}}{f_{m}} + 1} \right\rbrack}\left\lbrack \frac{Fk_{B}T}{P_{s}} \right\rbrack}.}}} & (16)\end{matrix}$

Eqn. (16) illustrates that sensitivity should be at a minimum formagnetic fields at frequencies f_(m) satisfying f_(c)<f_(m)<f_(L). Inthis region, both the signal and the noise scale as roughly 1/f_(m),resulting in an approximately flat frequency response. For themeasurements disclosed here, f_(c)=80 kHz and f_(L)=800 kHz, with thebest sensitivity between those two frequencies. At frequencies belowf_(c) or above f_(L), sensitivity is reduced due to various effects. Atfrequencies near or below f_(c), the flicker noise of the amplifier (aswell as other effects such as thermal drift of the ferrimagneticresonance or vibration) increases the oscillator phase noise relative tothe signal. For frequencies near or above the Leeson frequency f_(L),sensitivity may be compromised because the phase noise is independent off_(m) while the signal response decreases at 20 dB/decade as f_(m)increases.

If f_(m) additionally satisfies f_(c)<<f_(m)<<f_(L), Eqn. (16) may besimplified to

$\eta \approx {\frac{1}{2\sqrt{2}}\frac{\kappa_{L}}{\gamma}{\sqrt{\frac{Fk_{B}T}{P_{s}}}.}}$Setting F=1 in Eqn. (16) yields a sensitivity equivalent to that of anidealized (that is, with ideal amplifier and thermally noise limited),optimally coupled (κ₁=κ₂=κ₀/2, assuming an ideal amplifier) transmissioninterferometer.

This reflects what appear to be fundamental limits of oscillator phasenoise. Quantitatively, the Leeson effect dictates that

${{S_{\varphi}(f)} = {\left\lbrack {1 + \frac{f_{L}^{2}}{f^{2}}} \right\rbrack{S_{\psi}(f)}}},$where S_(ψ)(f) is the one-sided power spectral density of additive phaseshifts inside the oscillator loop, and S_(φ)(f) denotes the one-sidedpower spectral density of the oscillator's output phase noise. Asthermal noise produces a lower bound on S_(ψ)(f) and the effect of thisnoise on S_(φ)(f) is effectively enhanced for frequencies below f_(L),the intuition that a device can reach the naive thermal phase noiselimit (−177 dBm/Hz) for frequencies below f_(L) may be incorrect,regardless of any oscillator narrowing techniques that may be used. Thesame limits are observed in interferometric frequency discriminators.Ferrimagnetic Gradiometer

FIG. 10 shows a gradiometer 1000 that uses two ferrimagnetic oscillatormagnetometers 1002 a and 1002 b (collectively, magnetometers 1002) tomeasure the gradient of an external magnetic field. These ferrimagneticoscillator magnetometers 1002 a and 1002 b can be implemented asseparate devices, as in FIG. 10 , or as different ferrimagnetic crystalssubject to the same bias field from the same permanent magnet. In eithercase, the magnetometers' analog outputs 1003 a and 1003 b (collectively,analog signals 1003) (e.g., from the coupler 130 in FIG. 1A or the BJTcollector c in FIG. 6A) are mixed with an analog mixer 1004 to producean intermediate frequency (IF) signal that is digitized with a digitizer1006. As explained above, the magnetometers' analog outputs 1003 aremodulated with sidebands whose amplitudes are proportional to theapplied magnetic field. A magnetic field gradient across themagnetometers 1002 causes the magnetometers 1002 to produce outputs 1003with different sidebands. When mixed, these sidebands produce an IF beatproportional to the magnetic field gradient.

TABLE 1 Variables and Parameters for Ferrimagnetic OscillatorMagnetometers Name Symbol Approx. value Units Gyromagnetic ratio g_(e)≈2 unitless Bohr magneton μ_(B) 9.274 × 10⁻²⁴ J/T Vacuum permeability μ₀1.257 × 10⁻¹⁶ H/m Boltzmann constant k_(B) 1.381 × 10⁻²³ J/KGyromagnetic ratio γ ≈2π × 28 × 10⁹ rad · s⁻¹T⁻¹ System temperature TKelvin Amplifier wideband noise factor F unitless Amplifier flickernoise corner f_(c) Hz Amplifier input (sustaining) power P_(s) WIntrinsic linewidth κ₀ rad/s Input coupling rate κ₁ rad/s Outputcoupling rate κ₂ rad/s YIG resonant frequency ω_(y) rad/s MW drivefrequency ω_(d) rad/s Oscillator carrier frequency ω_(c) rad/sIntermediate frequency ω_(i) rad/s Loaded linewidth κ_(L) = κ₀ + κ₁ + κ₂rad/s Loaded quality factor Q_(L) = ω_(y)/κ_(L) unitless Unloadedquality factor Q₀ = ω_(y)/κ₀ unitless Leeson frequency f_(L) = κ_(L)/(2× 2π) Hz S-parameters S₁₁, S₁₂, S₂₁, S₂₂ unitless Total magnetic field B= B₀ + B_(sen) Tesla Bias magnetic field B₀ ≡ B₀{circumflex over (z)} TTest sensing magnetic field B_(sen) (t) T Test sensing magnetic fieldrms amplitude B_(sen) ^(rms) T Test sensing magnetic field ω_(m) =2πf_(m) rads/s angular frequency Test sensing magnetic field frequencyf_(m) = ω_(m)/(2π) Hz Demagnetization factors N_(x), N_(y), N_(z)unitless Saturation magnetization M_(s) A/m Single-sideband phase noise

(f_(m)) dBc/Hz power spectral density Single-sideband phase noisespectral density

^(1/2)(f_(m)) dBc/{square root over (Hz)} Time t s Oscillator voltagewaveform ν(t) V Oscillator voltage waveform amplitude V₀ V Oscillatorinstantaneous phase ϕ(t) rads Oscillator additive phase (noise orsignal) φ(t) rads Oscillator additive amplitude noise α(t) unitless Lineimpedance Z₀ Ohms Transformer turn ratios n₁, n₂ unitless Magneticsensitivity at frequency f_(m) η(f_(m)) T/{square root over (Hz)}Conclusion

While various inventive embodiments have been described and illustratedherein, those of ordinary skill in the art will readily envision avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications isdeemed to be within the scope of the inventive embodiments describedherein. More generally, those skilled in the art will readily appreciatethat all parameters, dimensions, materials, and configurations describedherein are meant to be exemplary and that the actual parameters,dimensions, materials, and/or configurations will depend upon thespecific application or applications for which the inventive teachingsis/are used. Those skilled in the art will recognize or be able toascertain, using no more than routine experimentation, many equivalentsto the specific inventive embodiments described herein. It is,therefore, to be understood that the foregoing embodiments are presentedby way of example only and that, within the scope of the appended claimsand equivalents thereto, inventive embodiments may be practicedotherwise than as specifically described and claimed. Inventiveembodiments of the present disclosure are directed to each individualfeature, system, article, material, kit, and/or method described herein.In addition, any combination of two or more such features, systems,articles, materials, kits, and/or methods, if such features, systems,articles, materials, kits, and/or methods are not mutually inconsistent,is included within the inventive scope of the present disclosure.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e., “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of” “only one of,” or“exactly one of.” “Consisting essentially of” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

The invention claimed is:
 1. A magnetometer comprising: a ferrimagnetic crystal comprising an ensemble of entangled electronic spins, the ensemble of entangled electronic spins having a resonance that shifts in response to an external magnetic field; a sustaining amplifier, in electrical communication with the ferrimagnetic crystal, to amplify a microwave signal modulated by a shift in the resonance of the ensemble of entangled electronic spins; and a digitizer, in electrical communication with the sustaining amplifier and/or the ferrimagnetic crystal, to digitize the microwave signal.
 2. The magnetometer of claim 1, wherein the ferrimagnetic crystal and the sustaining amplifier are connected in a transmission geometry.
 3. The magnetometer of claim 1, wherein the ferrimagnetic crystal and the sustaining amplifier are connected in a reflection geometry.
 4. The magnetometer of claim 1, wherein the sustaining amplifier comprises a bipolar junction transistor.
 5. The magnetometer of claim 1, further comprising: a bandpass filter, in electromagnetic communication with an input of the sustaining amplifier, to filter the microwave signal.
 6. The magnetometer of claim 1, further comprising: a bias magnet, in electromagnetic communication with the ferrimagnetic crystal, to apply a bias magnetic field to the ensemble of entangled electronic spins.
 7. The magnetometer of claim 1, wherein the shift in the resonance varies linearly with an amplitude of the external magnetic field.
 8. The magnetometer of claim 1, wherein the shift in the resonance modulates sidebands onto the microwave signal with amplitudes proportional to an amplitude of the external magnetic field at offset frequencies proportional to an oscillation frequency of the external magnetic field.
 9. The magnetometer of claim 8, wherein the magnetometer has a sensitivity versus the oscillation frequency of the external magnetic field that is substantially constant for f_(c)<f_(m)<f_(L), where is the oscillation frequency of the external magnetic field, f_(c) is an observed noise corner of the sustaining amplifier, and f_(L) is the Leeson frequency of the magnetometer.
 10. The magnetometer of claim 1, further comprising: an input coupling loop, inductively coupled to the ferrimagnetic crystal, to couple the microwave signal into the ferrimagnetic crystal; and an output coupling loop, inductively coupled to the ferrimagnetic crystal, to couple the microwave signal out of the ferrimagnetic crystal.
 11. The magnetometer of claim 10, further comprising: a directional coupler having an input port coupled to the output coupling loop, a through port coupled to an input of the sustaining amplifier, and a tap port coupled to the digitizer.
 12. The magnetometer of claim 1, further comprising: a feedback loop, in electromagnetic communication with the ferrimagnetic crystal and the sustaining amplifier, to generate and apply an error signal correcting an error between a frequency of the microwave signal and a center frequency of the resonance. 